vol. 179, no. 4
the american naturalist
april 2012
Causality and Persistence in Ecological Systems:
A Nonparametric Spectral Granger Causality Approach
Matteo Detto,1,2,* Annalisa Molini,3,4,5 Gabriel Katul,4,5 Paul Stoy,6 Sari Palmroth,4 and
Dennis Baldocchi2
1. Smithsonian Tropical Resource Institute, Apartado Postal 0843-03092 Balboa, Ancon, Panama; 2. Department of Environmental
Science, Policy, and Management, University of California, Berkeley, California 94720; 3. Masdar Institute Science and Technology, Abu
Dhabi, United Arab Emirates; 4. Nicholas School of the Environment, Duke University, Durham, North Carolina 27708;
5. Department of Civil and Environmental Engineering, Pratt School of Engineering, Duke University, Durham, North Carolina 27708;
6. Department of Land Resources and Environmental Sciences, Montana State University, Bozeman, Montana 59717
Submitted December 20, 2010; Accepted December 12, 2011; Electronically published February 20, 2012
Online enhancements: appendixes, zip file.
abstract: Directionality in coupling, defined as the linkage relating
causes to their effects at a later time, can be used to explain the core
dynamics of ecological systems by untangling direct and feedback
relationships between the different components of the systems. Inferring causality from measured ecological variables sampled through
time remains a formidable challenge further made difficult by the
action of periodic drivers overlapping the natural dynamics of the
system. Periodicity in the drivers can often mask the self-sustained
oscillations originating from the autonomous dynamics. While linear
and direct causal relationships are commonly addressed in the time
domain, using the well-established machinery of Granger causality
(G-causality), the presence of periodic forcing requires frequencybased statistics (e.g., the Fourier transform), able to distinguish coupling induced by oscillations in external drivers from genuine endogenous interactions. Recent nonparametric spectral extensions of
G-causality to the frequency domain pave the way for the scale-byscale decomposition of causality, which can improve our ability to
link oscillatory behaviors of ecological networks to causal mechanisms. The performance of both spectral G-causality and its conditional extension for multivariate systems is explored in quantifying
causal interactions within ecological networks. Through two case
studies involving synthetic and actual time series, it is demonstrated
that conditional G-causality outperforms standard G-causality in
identifying causal links and their concomitant timescales.
Keywords: spectral Granger causality, complex systems, periodic forcing, photosynthesis-respiration coupling.
Introduction
Systems exhibiting mutualisms and competition or other
forms of two-way interactions are ubiquitous in ecology.
Some examples taken from population and community
* Corresponding author; e-mail: dettom@si.edu.
Am. Nat. 2012. Vol. 179, pp. 524–535. 䉷 2012 by The University of Chicago.
0003-0147/2012/17904-52705$15.00. All rights reserved.
DOI: 10.1086/664628
ecology include Lotka-Volterra predator-prey systems
(Roughgarden 1975 and references therein; Chesson 1994;
Holmes et al. 1994), nutrient recycling (Loreau 2001), and
human-biosphere interactions (e.g., Brander and Taylor
1998; Raupach 2007). Some populations, such as insects,
that display nonlinear demographic dynamics (Costantino
et al. 1997) exhibit complex endogenous behaviors, even
in the absence of environmental variability. For others, the
interplay with environmental drivers is crucial in determining their physical state, as in the case of vegetation
and soil moisture in water-limited regions (Detto et al.
2006).
Common to all of these examples is some understanding
of the dynamical nature of the core ecological system,
which is often encoded in a set of autonomous differential
equations (e.g., a Lotka-Volterra-like system). These systems are causal in that their dynamics can be explained
in terms of conditionals in the form of “if A had not
occurred, B would not occur.” When such dynamics lead
to self-sustained oscillations and, at the same time, some
external time-dependent forcing is acting on this system,
it becomes difficult to distinguish the signature of autonomous oscillatory dynamics from the effects of the external
forcing. In a similar context, any coupling among ecological variables induced by external drivers is hard to distinguish from the endogenous correlation structure of the
core ecological system. Consequently, it is not surprising
that in ecological time series analysis, one of the most
challenging goals is to determine how one state variable
causes the future evolution of another.
The standard approach to confront this challenge is
through controlled manipulative experiments. This empirical approach provides strong evidence because it can
reveal unambiguous mechanistic causal relationships.
However, in many natural settings, such manipulative ex-
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Granger Causality in Ecological Systems 525
periments are difficult to implement. For example, experimental manipulation of both air temperature and CO2
concentration are difficult at the ecosystem scale (Shaw et
al. 2002), and findings from such experiments may be scale
dependent or constrained by the limited spatial scale. Even
well-designed manipulative experiments are not free from
spurious results, as may occur when covariates are inadvertently changed by the manipulation.
Another less direct approach to infer causal interactions
(if possible) relies on observational data (Shipley 2000),
for example, using structural equation modeling (SEM).
Path analysis, a special case of SEM, can be formulated as
a multiple regression problem aimed at decomposing different sources of correlation within an ecological system
rather than explicitly accounting for directional coupling
in time (i.e., causes preceding effects). However, the existence of correlation itself does not necessarily entail causation as implied by an action and a subsequent reaction.
Correlation metrics are essentially symmetric and cannot,
by definition, measure time asymmetry—often considered
the signature of causality (Hlavackova-Schindler et al.
2007; Lungarella et al. 2007; Paluš and Vejmelka 2007)—
nor can they distinguish between unidirectional and bidirectional couplings. Additionally, path analysis handles
with difficulty any colinearity (Petraitis et al. 1996).
The idea of causality has been developed and studied
in a number of disciplines ranging from philosophy to
science and by no means can be covered in a single article
(e.g., see Peters 2001, pp. 128–136, for a comprehensive
synthesis of definitions and measures in ecology and Pearl
2009 for applications in statistical inference). The scope
of this work is a specific yet widely used type of causality
known as Granger causality (or G-causality; Granger
1969), where cause and effect are linearly related. This
causality metric originated in econometrics but is now
proliferating to a number of disciplines, including geosciences (Salvucci et al. 2002; Kaufmann et al. 2008; Smirnov and Mokhov 2009) and neurosciences (Bernasconi
and König 1999; Ding et al. 2006). G-causality explains
phenomena by showing them to be due to effects originating from prior causes in time. When these prior causes
are accounted for, predictions of the phenomena are improved against a null hypothesis that does not account for
these prior causes. Hence, the premise of G-causality encodes two of the three cardinal aspects of causality (explanation, prediction, and teleology) singled out by Mayr
(1960) in biology. However, due to the predominantly
periodic nature of external drivers encountered in ecology,
as well as the complex nature of ecological systems, an
extension of G-causality to the frequency domain (i.e., a
spectrally transformed series) is warranted and proposed
here. The basic idea is to decompose and analyze causal
relationships as a function of the frequency at which they
occur instead of as a function of time.
Spectral methods have been widely adopted in ecology
and are reviewed elsewhere (e.g., Platt and Denman 1975;
Priestley 1981). In the multitude of possible ecological
applications, the goal remains the same: to move from a
framework where variability of the analyzed system is a
function of time (time domain) to a new approach where
the variance can be decomposed scale by scale, allowing
for a clear view of how different frequencies impact the
considered process.
The principle of causation (Mayr 1960) and the search
for causality in relation to space and timescales within an
ecological process have a long history in ecology, preceding
even the work by Granger in economics. Kershaw’s (1963)
seminal work investigated the length scales of vegetation
clumping while trying to assess causality by linking the
scale of the pattern with its generating mechanism. In this
case, short-length scale patterns (∼10 cm) suggested a
causal connection based on plant morphology, while
length scales of tens of meters implied interactions with
topography, water drainage networks, or soil composition
(i.e., mainly but not completely exogenous mechanisms).
A similar approach can be applied to ecological systems
displaying a linear causal structure in time. In a linear
system, an exogenous forcing at a given period propagates
through the system unaltered (except through its phase
angle) to produce an effect at the same frequency.
The approach may fail in the presence of nonlinearity.
In this case, a driver acting on a specific frequency can
produce a response with variability at one or more different frequencies (Pascual and Ellner 2000). Extracting
information from nonlinear time series is inherently challenging: “when the analyzed data cannot be distinguished
from realizations of a linear process, it is hardly expected
to extract useful information by a nonlinear descriptor”
(Paluš 2008, p. 320), especially for short, noisy time series.
In any case, linearized models can often provide acceptable
approximations of the dynamics of ecological systems, allowing for the estimation of main direct interactions by
simpler linear causal metrics (Nisbet and Gurney 1982;
Royama 1992). In spite of this, nonlinearity tests may establish the extent to which linear analysis is pertinent to
ecological data without generating spurious causal relationships (Paluš 2008; Papana et al. 2011).
Here, the main theory describing G-causality in the time
domain is briefly reviewed. Next, the analysis is extended
to the spectral domain, where systems of reciprocally interacting variables are subjected to stochastic forcing. Finally, a novel approach based on conditional G-causality
in the spectral domain is presented. This conditional statistic, hereafter referred to as conditional spectral G-causality, allows for differentiating direct causal linkage from
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526 The American Naturalist
indirect causal linkage between multiple state variables
even under the action of external drivers. This approach
builds on the so-called spectral decomposition of G-causality (Chen et al. 2006; Dhamala et al. 2008), where time
series are transformed to the frequency domain and the
causal relationships between periodic drivers and self-sustained oscillations are assessed scale by scale.
As an illustration, two applications of G-causality are
presented: one theoretical (simulation based) and the other
empirical. The theoretical case focuses on a system composed of primary producers and decomposers that are
allowed to interact in a compartmental model while incorporating stochastic components. The empirical case explores the causal interaction between photosynthesis and
soil respiration in forested ecosystems when time series
measurements are conditioned on soil temperature and
the presence of leaves. This empirical case study was previously analyzed by Stoy et al. (2007), using a lagged correlation, or “pulse response,” analysis. Their analysis failed
to establish a strong causal relationship precisely because
periodicity occurred at multiple timescales (in the temperature and incident radiation) overlapping with the
characteristic scale of endogenous interaction between
photosynthesis and respiration. These two applications are
intended to provide a methodological basis for the detection of G-causality in ecological systems while illustrating
its usefulness as a diagnostic approach.
Methods of Analysis
Granger Causality: A Brief Review
Granger (1969, p. 430) defines causality as follows: “if
some other series yt contains information in past terms
that helps in the prediction of xt and if this information
is contained in no other series used in the predictor, then
yt is said to cause xt.” This connection between causality
and predictability led Granger to express linear causality
in a parametric form, drawing from established tools in
time series analysis (see Box and Jenkins 1970), such as
autoregressive models.
If an autoregressive representation of order m is used,
the bivariate model assessing the influence of a random
variable y on another random variable x (and vice versa)
is given by
冘
冘
m
xn p
jp1
jp1
a 2, j yn⫺j ⫹ ␧n ,
(1a)
b2, j yn⫺j ⫹ yn ,
(1b)
jp1
m
yn p
冘
冘
m
a 1, j x n⫺j ⫹
m
b1, j x n⫺j ⫹
jp1
where ␧ and y are the prediction errors while a and b are
coefficients describing the linear interactions between the
variables, with subscript j indicating time lags. When the
above equation is compared with a univariate model,
m
x n p 冘jp1 a j x n⫺j ⫹ hn, and when the multivariate model
outperforms the univariate case (e.g., j␧2 ! jh2), y is said to
have a causal effect on x (and similarly for the effect of x
on y). This is the statistical interpretation of causality proposed by Granger (1969) and is commonly referred to as
Granger causality, or G-causality.
Note here that correlation itself does not necessarily
imply an improvement in prediction (Aldrich 1995). Correlation is a measure of coupling strength, which can originate from both causation and dependence on common
causes.
G-causality is a measure of coupling, with time directionality being explicit. For this reason, it is based on prediction errors rather than on linear interactions among
coefficients. Traditionally, it is expressed as the ratio between the residual variance of the bivariate and the null
model (i.e., the univariate case) and is given as
Gyrx p ln
jh2
.
j␧2
(2)
If the variables x and y do not interact, there will be no
improvement in using y to predict x; that is, j␧2 ≈ jh2 and
Gyrx ≈ 0, even if the two variables are correlated. If otherwise y has a causal influence on x, j␧2 ! jh2, so Gyrx 1
0. The causal effect of x on y, Gxry is defined in a similar
manner. G-causality varies between 0 and infinity; it is
analogous but not identical to ⫺ ln (1 ⫺ R 2), where R2 is
the usual coefficient of determination (the proportion of
the variance explained by a linear model). Because of the
temporal sequence shown in equations (1), it is clear that
G-causality can capture only functional relationships for
which cause and effect are sufficiently separated in time.
From Time to Frequency: Spectral G-Causality
and the Nonparametric Approach
Ecological processes, especially those connected with the
exchanges of mass and energy between biosphere and
atmosphere, often display strong oscillations due to diurnal or seasonal cycles (Baldocchi et al. 2001b; Katul et
al. 2001). Interannual variability in many natural ecosystems is regulated by large climatic fluctuations (Stenseth
et al. 2002), which can be revealed by spectral analysis.
Hence, it is convenient to extend the definition of the
causality in equation (2) to the frequency domain and
introduce the spectral analogue of G-causality I as a function of frequency f (Geweke 1982), using
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Granger Causality in Ecological Systems 527
{
}
Sxx(f )
I(f )yrx p ln
,
Sxx(f ) ⫺ [Gyy ⫺ (Gxy2 /Gxx )]FH xy(f )F2
(3)
where Sxx(f ) is the value of the spectral density of x (also
called the periodogram, or power spectrum) at frequency
f, G is the error covariance matrix for the system of equations (1) and H(f ) is a spectral transfer function matrix
resulting from rewriting system (1) in the Fourier domain.
As in equation (2), if the variables x and y are not
interacting, the numerator and denominator of equation
(3) are equal and I(f )yrx ≈ 0. If y manifests a causal influence on x at a specific frequency fs, then I(fs)yrx 1 0.
The robustness of these statistics must be tested against
the null hypothesis that the two series do not exhibit any
causal interactions at a particular frequency. In general,
the analytical solution for the causality structure of the
system is not a priori known, necessitating the use of
Monte Carlo simulation methods and surrogate data. The
null hypothesis that no linear causal relationships exist
among the variables considered is here tested using the
so-called iterative amplitude adjusted Fourier transform
(IAAFT) surrogates. The IAAFT surrogates, recently
adopted in testing rainfall nonlinearity and cross-scale causality in time (Roux et al. 2009; Molini and Katul 2010),
simultaneously preserve the probability distribution and
the power spectrum of the original time series. Thus, the
linear correlation structure of the original time series is
conserved by the IAAFT, while any other form of coupling
or nonlinear correlation is destroyed.
Since S, H, and G can be inferred from the time series,
equation (3) does not require any assumption regarding
the autoregressive order of the model representing the data
and the determination of I(f ) can be considered nonparametric. The spectral density matrix S can be estimated
directly from the spectral and cross-spectral functions (using Fourier, wavelet transform, or analogous spectral
methods). The matrix factorization (Wilson 1972; Dhamala et al. 2008) allows H and G to be computed from the
equality S p HGH , where the prime indicates the matrix
adjoint. Full details of such factorization are provided in
appendix A, available online.
This approach is particularly effective for time series
exhibiting apparent periodicities, superimposed on longrange memory, which would otherwise require higherorder autoregressive models to reconstruct the observed
dynamics. The estimation error of an autoregressive model
is known to increase with order. Moreover, order misspecifications may produce spurious causality effects (Hlavackova-Schindler et al. 2007).
The adopted scheme can be extended to the multivariate
case, which now involves k stochastic variables (z1, ..., zk).
In this extension, it becomes possible to compute the so-
called conditional G-causality I(f )yrxFz1, …, zk (see details in
app. A); that is, y causes x, given that z1, ..., zk cause x or
y (Chen 2006).
Statistics Performance and Key Applications
Case 1: A Stoichiometric Model
The interactions between primary producers and decomposers are often studied by using a stoichiometric approach. Following Daufresne and Loreau (2001), the cycle
of organic and inorganic matter within an ecosystem can
be represented by three compartments: primary producers
(P), detritus (L), and decomposers (D), along with a pool
of inorganic nutrients (N).
The detritus generated by producers constitutes the substrate for the communities of decomposers, which mineralize nutrients and release carbon back into the atmosphere. All the compartments have constant C : N ratios
(here denoted by a for P and L and by b for D, with
a 1 b), which allows formulating explicit mass-balance
constraints and expressing stocks and fluxes independently
in terms of C or N. The models combine indirect mutualism and competition because decomposers recycle nutrients as well as compete with producers for them. The
dynamical equations for a closed system (zero dilution
rate) expressed in nutrient stocks are
Ṗ p Fn ⫺ eP,
L̇ p eP ⫺ Fd ,
(4)
Ḋ p Fi ⫹ Fd ⫺ mD,
Ṅ p mD ⫺ Fi ⫺ Fn ,
where e and m are the primary producers’ death rate and
the decomposers’ mineralization rate, respectively, while
Fn, Fd, and Fi represent the nutrient uptake, decomposition, and immobilization fluxes, respectively. These
fluxes can be described using a nutrition consumption
function of a Lotka-Volterra type (nonlinear interaction)
or, alternatively, using a donor-controlled mechanism (linear interactions). For this model, the results of the donorcontrolled mechanism approach were shown to be qualitatively the same as the Lotka-Volterra type (Daufresne
and Loreau 2001). Namely, the system can be assumed
dynamically linear. This schematization is feasible only
when decomposers are carbon limited. The immobilization flux can be expressed as Fi p [(a ⫺ b)/b]Fd. If a
donor-controlled interaction type is assumed, nutrition
and decomposition fluxes are Fn p nN and Fd p dL, respectively, where n is the plant nutrition rate and d is the
detritus decomposition rate. Because the total nutrient
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528 The American Naturalist
Figure 1: Sample realization of normalized nutrient stocks for primary producers and decomposers, simulated with equation (5) (P* p
N* p 0.25NT).
stock is constant for a closed system (NT p N ⫹ P ⫹
L ⫹ D), equation (4) reduces to three independent equations:
Ṗ p nNT ⫺ (e ⫹ n)P ⫺ nL ⫺ nD,
L̇ p eP ⫹ dL,
Ḋ p
(5)
a
dL ⫺ mD.
b
The stable steady state, obtained by setting all time derivatives to 0, is given by
P* p
NT
,
x
e
L * p P *,
d
(6)
ae *
D* p
P ,
bm
e
N * p P *,
n
where x p 1 ⫹ (e/n) ⫹ (e/d) ⫹ (a/b)(e/m).
In the presence of environmental noise, the discrete
form of the deterministic system in equation (5) is analogous to the linear first-order autoregressive model in
equations (1). The introduction of random terms is assumed to represent the variability of the nutrition, decomposition, mineralization, and mortality rates as may
be induced by hurricane, pest or disease, changes in mineralization rate in response to temperature fluctuations,
and so on.
In natural (not controlled) ecosystems P, L, D, and N
vary over very different timescales (from years to days)
and are characterized by different abundances in turn influencing parameters m, e, d, and n. We simulated 50 short
realizations, each with 100 weekly steps, to mimic real
ecological time series, for the system in equation (5). For
simplicity, the total nutrient stock is equally divided at
equilibrium among organic and inorganic compartments
(P * p L* p D * p N * p NT/4) and a p 1, b p 0.8,
n p e p d p 0.5, m p 0.625. The three stochastic variables are assumed to be independent Gaussian noises with
variance equal to 10⫺3. Note that this assumption is not
required and analogous results would be obtained with a
different type of noise (e.g., Poisson) even if the noises
were not independent (e.g., autocorrelated or correlated
with each other, as in natural ecosystems). As an illustration, figure 1 shows one realization for primary producers
and decomposers, expressed in stock of nutrients, while
figure 2 depicts the conditional G-causality for different
interaction schemes among L, P, and D.
From these simulations, the conditional G-causality correctly detected the dynamical interactions represented in
the model; L and D have a direct influence on P (fig. 2b,
2c), P controls L (fig. 2d), but D has no direct effect on
L (fig. 2e). Finally, D is controlled by the amount of detritus
L (fig. 2h), as the decomposers are carbon limited. Pairwise
G-causalities in figure 2 (dashed lines) fail to determine
unambiguously the real direct interactions and their
strengths. In fact, the G-causalities in figure 2f and 2g are
not 0. Producers, for example, have an effect on decomposers but only indirectly, as mediated by the detritus pool
(fig. 2g). The bivariate causality of L on P (fig. 2b) is also
overestimated. The detritus appears to have more influence
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Granger Causality in Ecological Systems 529
cause
P
-3
3
x 10
L
a)
D
b)
1
c)
1
2
P
0.5
1
0
0.5
0
0
-3
d)
effect
1
3
x 10
e)
f)
1
2
L
0.5
0.5
1
0
0
D
L| P
D
L
0
-3
g)
1
h)
1
3
x 10
i)
2
D
0.5
0
0.5
0
0.1
0.2
0.3
0.4
0
1
0
0.1
0.2
0.3
0.4
0
0
0.1
0.2
0.3
0.4
-1
frequency (time )
Figure 2: Conditional Granger causalities (G-causalities; bold solid lines) for the model simulations indicate the direct interactions among
two variables, given the effect of the third (e.g., the effect of L on P, given D; b). Spectral estimations are shown in the diagonal panels,
and the G-causalities for the bivariate case are shown as dashed lines. The bold lines represent the desired goal of the analysis because they
isolate direct effects, while the dashed lines indicate the apparent connections that are due to a combination of direct and indirect effects.
Legend in f applies to all panels.
on producers than on decomposers (cf. fig. 2b, fig. 2h)
when actually the model parameters were originally selected to show the opposite (n p 0.5, a/bd p 0.625).
The choice of parameters determines the frequency corresponding to the peak in the power spectra (fig. 2, diagonal panels), which in this example is fmax p 0.13
week⫺1.
The performance of the G-causality in detecting real
interactions was tested by repeating this numerical experiment 1,000 times using 50 and 10 realizations separately
for each simulation. The intent of these different realization sizes is to assess the rapid convergence of the Gcausality metric. Because the model structure is known a
priori here, analytical solutions can be obtained directly
from the model’s parameters (Chen et al. 2006). In figure
3, a case of interest is shown that represents the decomposers’ carbon limitation hypothesis, that is, the influence
of the detritus on decomposers after taking into account
the effect of producers. When the 50 realizations are used,
estimates that are much closer to the analytical solution
are obtained when compared to using only 10 realizations.
A large number of numerical simulations suggest that
unless the variance of the noise becomes too small compared to measuring errors, conditional G-causality always
detected the correct directionality in this system. This finding was not altered even when the noises were correlated
with each other. For further examples, see appendix B,
available online; original Matlab code developed and used
in the examples is also available online as a zip file.
Case 2: Ecosystem Application
Based on manipulation experiments, stable and radio isotope techniques, and soil gas phase CO2 concentration
measurements, a number of studies (Horwath et al. 1994;
Andrews et al. 1999; Hogberg et al. 2001) have suggested
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530 The American Naturalist
Figure 3: Analytical conditional Granger causality for the influence of the detritus on decomposers, given the effect of producers (carbon
limitation hypothesis). The 95% envelopes are obtained from 1,000 simulations of the model with 50 realizations (dashed lines) and 10
realizations (dotted lines).
that canopy assimilation can modulate soil respiration (i.e.,
a causal link exists between these two variables) with a lag
time that varies from a few hours to several days (see
Kuzyakov and Gavrichkova 2010 for a review). Representing the relationship between carbon inputs and outputs in ecological models is critical to understand ecosystem carbon cycling. However, detecting causal links
between photosynthesis and respiration is complicated by
the interacting role of abiotic factors (temperature and
water availability), as well as physical and biological CO2
transport mechanisms. The diversity of primary producers
and decomposers implies the coexistence of several compartments with different turnover rates (Loreau 2001),
some of which may act in synchrony with these physical
drivers.
Stoy et al. (2007) conducted a lagged cross-correlation
analysis (the correlation among variables at different offset
time lags) between canopy photosynthesis (Ac) estimated
from eddy-covariance time series and forest floor respiration (Rs) estimated from automated chamber time series.
The study was conducted in adjacent planted pine (PP)
and hardwood (HW) forested ecosystems within the Duke
Forest, Durham, North Carolina. A statistically significant
peak in the cross-correlation function between Ac and Rs
emerged at time lags between 1 and 3 days in both forest
stands during summer months under finite canopy photosynthesis. The time lag was consistent with timescale
arguments that consider the soil transport of CO2 as strictly
diffusive and adds a representative phloem transport time-
scale (the time required for carbon to travel from leaf to
phloem to root or endomycorrhizal surface) to complete
the pathway from leaf to soil to atmosphere.
Stable isotopic studies also revealed the signature of
photosynthesis in soil respiration in PP at timescales of
less than 1 week (Andrews et al. 1999). Separate studies
have shown that most soil-respired carbon entered the
ecosystem via photosynthesis no more than 1 month earlier (Taneva et al. 2006; Drake et al. 2008). Collectively,
these independent studies demonstrate causal links between canopy photosynthesis and root respiration (i.e.,
delayed response following a carbon input) on short
timescales.
When the correlation analysis in Stoy et al. (2007) was
repeated in the winter months and the hardwood forest
had minimal leaf area and negligible photosynthesis, apart
from the presence of some evergreen species in the footprint of the tower, surprisingly similar time lags for the
maximum cross-correlation between Ac and Rs emerged.
It was concluded that the observed wintertime lag is due
to multiple forcing functions (mainly radiation and soil
temperature) rather than carbon input into the soil. When
the effects of soil temperature on respiration were partially
filtered (via a nonlinear filter), the maximum cross-correlation between Ac and Rs within the hardwood forest was
weaker yet still maintained statistical significance. These
findings motivate the development of a robust test of causality at multiple timescales where ecosystem CO2 assimilation, transport, and respiration occur.
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Granger Causality in Ecological Systems 531
To illustrate the advantages of conditional G-causality
in interpreting the input-output dynamics of the photosynthesis–soil respiration system, the same published data
set by Stoy et al. (2007) was analyzed. The study comprises
4 years of automated chamber measurements of hourly
forest floor fluxes. These measurements were taken at different locations within the stand, along with eddycovariance measurements of water vapor and CO2 fluxes
above the forest canopy at PP and HW (for a more exhaustive description of the experiment, see also Palmroth
et al. 2005; Drake et al. 2008).
The time series of Rs, Ac, and soil temperature (Ts) for
each stand were divided into blocks of 30 days. The variables for each block were normalized to have zero mean
and unit variance. The ensemble spectra and cospectra
were then evaluated as averages in the growing season
series (April–August) and non–growing season series (October–February). The analysis was carried out separately
for these two periods to demonstrate the causal role of Ac
on Rs, which should hold true only when leaves are present.
Figures 4 and 5 represent the results of spectral G-causality analysis for PP and HW, respectively. The ensemble
spectra show a consistent peak corresponding to the daily
cycle. At the same frequency, the bivariate G-causalities
are all significant. Reference to the causality of Rs on Ac
is also shown to be 0 (respiration has no influence on
assimilation at these scales). When conditional G-causality
is computed (conditioned on Ts), the peak at 1 day⫺1 is
greatly reduced, particularly in the wintertime, when the
deciduous forest is dormant.
This analysis clearly demonstrates that conventional
correlation analysis or standard G-causality can fail in isolating causal linkages in multivariate systems, while conditional G-causality agrees with logical expectations derived from other experimental techniques such as stable
and radio isotopes. In winter, there is little carbon input
from a near-leafless HW stand, yet forest floor respiration
remains appreciable, primarily due to heterotrophic and
autotrophic soil respiration. After conditioning on soil
temperature with the conditional G-causality approach, it
becomes clear that much of this respiration is not induced
by photosynthesis but by Ts, which oscillates on diurnal
frequencies. Significance levels were obtained using IAAFT
data of Ac series when evaluating G-causality and its conditional counterpart.
Because Ac is derived by flux separation of net ecosystem
spectrum
PP - SUMMER
PP - WINTER
0.1
Rs
Rs
Ac
Ac
T
T
s
s
0.05
0
G-causality
0.3
0.2
Ac
Rs
Ac
Rs
Ac
Rs
Ac
Ac
Rs | Ts
Rs
Ac
Rs | Ts
0.1
0
0.5
1
1.5
frequency [day-1]
2
0.5
1
1.5
2
frequency [day-1]
Figure 4: Granger causality (G-causality) and conditional G-causality (conditioned on soil temperature Ts) between soil respiration (Rs)
and canopy assimilation (Ac) rate for the evergreen pine plantation (PP) are shown in the bottom panels, respectively. The ensemble spectra
for the respective summer and winter series are shown in the top panels for reference. Dashed lines represent 95% confidence intervals
around the null hypothesis (no G-causality), obtained via an ensemble of 1,000 iterative amplitude adjusted Fourier transform surrogate
time series. Note that the bivariate G-causality overestimates the interaction between Ac and Rs when temperature is not taken into account.
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532 The American Naturalist
spectrum
HW - SUMMER
HW - WINTER
0.1
Rs
Rs
Ac
Ac
T
T
s
s
0.05
0
G-causality
0.8
0.6
0.4
Ac
Rs
Ac
Rs
Rs
Ac
R
Ac
Ac
s
Ac
Rs | Ts
Rs | Ts
0.2
0
0.5
1
1.5
-1
frequency [day ]
2
0.5
1
1.5
2
-1
frequency [day ]
Figure 5: Same as figure 4 but for the broadleaf deciduous (hardwood [HW]) forest adjacent to the evergreen pine plantation. Note the
reduced peak in the conditional Granger causality (G-causality) at 1 day⫺1 during winter months, compared to summer periods, or when
compared to standard pairwise G-causality.
CO2 exchange (Stoy et al. 2006), artificial correlations between Ac and ecosystem respiration may bring undesirable
effects into their correlation even though the two measurement systems are independent (eddy covariance and
automated chambers). In principle, this is not an issue for
directional causality because self-correlation should manifest itself only in the interdependence (instantaneous causality). However, to exclude any doubts, the same analysis
was repeated using a canopy conductance model with bulk
conductance obtained independently using water vapor
flux measurements from the eddy-covariance systems and
vapor pressure deficit observations. Because photosynthesis is tightly coupled with bulk conductance, and assuming
there is minor variation in the effective ratio of intercellular to ambient atmospheric CO2, then variations in bulk
conductance can be used as surrogates for variation in Ac
without resorting to any estimates of ecosystem respiration. The results were similar to those derived using the
eddy-covariance CO2 flux data for Ac (not shown) even
though the causalities were slightly reduced. This further
confirms the suitability of spectral conditional G-causality
while supporting the hypothesis that relationships between
inputs and outputs of carbon are causal at short timescales.
Drivers for Ac, namely, photosynthetically active radiation, generally exhibit more temporal variability than soil
temperature on such short timescales. Carbon flows within
the plant and into the soil are presumably dampened (and
lagged) by the presence of plant carbon pools. Nonetheless,
traces of variability connected with the plant activity can
fingerprint the soil processes, as evidenced in this study
as well as other independent stable isotope analysis.
Discussion and Conclusions
Determining the arrow of causal linkages between two or
more time-varying measures is now receiving significant
attention in a number of disciplines such as economics,
neuroscience, and climate science. The dynamics of many
ecological systems share a number of striking similarities
with these fields (marked variability over a wide range of
timescales and space scales, periodic forcing imposing
strong correlation but not causation among variables, etc.).
Causality, as defined in Granger’s (1969, 1988) work, embodies the dependence between two time series with a
nonzero lag time in their cross-correlation. A spectral extension of G-causality was proposed here to determine the
scale-dependent interactions among ecological variables.
This spectral approach presents a decisive advantage when
compared with the time-based scheme since oscillatory
forcing can be identified, localized within a specific and
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Granger Causality in Ecological Systems 533
limited number of frequencies (e.g., seasonal, diurnal, interannual, etc.), and removed via conditional analysis. Despite these advantages, the spectral G-causality and the
conditional G-causality remain underexploited tools in
ecological systems (whether in models or in data). The
main motivation here was to illustrate their potentials in
untangling and explaining causal relationships between the
different components of ecological systems.
Broadly speaking, persistence, long-term memory, or
long-range dependence often characterizes ecosystem processes. This memory is encoded in the measured autocorrelation function, in the spectral density function, or
in the coefficient matrix of an autoregressive process, depending on whether a parametric or a nonparametric version of G-causality is adopted. Soil respiration, like other
processes in the soil system, “remembers” the characteristics of the past states. Changes in soil states are slow
when compared to the aboveground system, as the processes are mediated by storage in the plant pools, diffusion
in the soil-plant system, and the complex bonding structure of soil organic matter. Respiration depends not only
on instantaneous combinations of soil temperature and
soil moisture but also on the complex interactions between
carbon pools (e.g., standing biomass and microbial biomass). At the Duke Forest sites, both bacteria and fungi
are actively present. If this memory, which is clearly beyond the diurnal cycle, is persistent enough, traces of these
interactions may be revealed via conditional G-causality.
This topic is currently under investigation, and fortunately,
the time is ripe for stepping into long-term analysis of
ecological systems, given the wealth of ecological measurements now available from the FLUXNET network
(Baldocchi et al. 2001a) and other similar databases. This
method, as any other statistical method, does not provide
any mechanistic explanation of the origin of the interactions. However, it is an invaluable tool that can be exploited to verify ecological hypotheses. Future development of this type of analysis may begin to incorporate
some nonlinear causal interactions (e.g., Marinazzo et al.
2008), namely, the situation in which a forcing at a given
frequency produces an effect over one or more different
frequencies.
Acknowledgments
We thank M. Dhamala for useful discussions. The data
collection at the Duke Forest was supported by the U.S.
Department of Energy through the Office of Biological
and Environmental Research Terrestrial Carbon Processes
program (Free-Air CO2 Enrichment and National Institute
for Climatic Change Research grants DE-FG0295ER62083, DE-FC02-06ER64156). M.D. and D.B. ac-
knowledge funding from the National Science Foundation
(grant 0628720). G.K. and A.M. acknowledge funding
from the National Science Foundation (grants NSF-EAR10-13339, NSF-CBET-10-3347, NSF-AGS-11-02227).
M.D. also acknowledges the Global Forest Carbon Research Initiative at the Center for Tropical Forest Science.
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Associate Editor: Benjamin M. Bolker
Editor: Ruth G. Shaw
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